Self-complementary Graphs

Abstract

Samokomplementarni grafovi su zanimljivi jer čine beskonačnu klasu grafova i imaju jaka strukturna svojstva. Na primjer, samokomplementaran graf mora imati točno $\frac{n(n-1)}{4}$ bridova, radijus 2, dijametar 2 ili 3 i oni postoje za sve izvodive n. U radu su predstavljeni rezultati brojnih matematičara koji su proučavali samokomplementarne grafove u proteklih 50 godina. Vidjeli smo da su neki od njih korisniji pri dokazivanju da graf nije samokomplementaran. Zapravo, ne postoji jednostavan način kojim bismo dokazali da je neki graf samokomplementaran. Kod ovakvih grafova problem predstavalja ne samo njihovo prepoznavanje, nego općenito brojnost i međusobna izomorfnost.Self-complementary graphs are interesting because they form an infnite class of graphs and have strong structural properties. For example, self-complementary graphs must have exactly $\frac{n(n-1)}{4}$ edges, radius 2 and diameter 2 or 3 and they exist for every feasible value n. In this paper we present results discovered by the mathematicians who studied self-complementary graphs during the last 50 years. We have shown that some of them are more useful in proving that some graph is not self-complementary rather than it is self-complementary. In fact, there is no an easy way to prove that graph is self-complementary. The problem is not just in recognision of those graphs, but also in their number and mutual isomorphism